Manifolds with Kinks

• Manifolds with kinks are spaces with singular boundaries that include smooth interiors, edges, corners, and cusps, unifying diverse geometric structures.
• The framework uses inward sectors—cones of admissible tangent directions—to derive precise asymptotic expansions for Gaussian-kernel graph Laplacians.
• In the Lorentzian setting, submanifolds-with-kinks support piecewise-smooth spacelike foliations, ensuring well-posed Bohmian trajectories via flux continuity.

Searching arXiv for the cited papers and closely related work on manifolds with kinks. Manifolds with kinks are a class of manifolds with possibly singular boundary that contains smooth manifolds without boundary, smooth manifolds with boundary, manifolds with corners, and more singular spaces such as pyramids, cones, and certain cusps. In the recent graph-Laplacian literature, the purpose of the notion is to provide a single geometric framework in which interior points, smooth boundary points, corners, and more singular boundary-like structures admit a common asymptotic analysis for Gaussian-kernel operators. In a distinct Lorentzian usage, submanifolds-with-kinks are stratified submanifolds in which at most two smooth pieces meet along a codimension-1 kink set; this framework is used to formulate piecewise-smooth spacelike foliations while preserving the well-posedness and equivariance of Bohmian trajectories (Pal et al., 10 Jul 2025, Struyve et al., 2013).

1. Definitions and scope

A topological $d$-manifold with kinks $M$ is a paracompact, Hausdorff space covered by charts of two types. Interior charts $(U,\phi)$ have $\phi:U\to\mathbb{R}^d$ a homeomorphism onto an open set. Border charts $(U,\phi)$ have $\phi:\overline U\to\mathbb{R}^d$ a homeomorphism onto the closure of an open set whose boundary has at least ${\mathcal C}^0$-regularity. Two charts are ${\mathcal C}^k$-compatible if their transition $\psi\circ\phi^{-1}$ extends to a ${\mathcal C}^k$-diffeomorphism of ambient open sets in $M$0. A smooth manifold with kinks is one admitting a maximal atlas of $M$1-compatible interior or border charts (Pal et al., 10 Jul 2025).

This definition is explicitly broad. It contains smooth manifolds without boundary, smooth manifolds with boundary, manifolds with corners, and more singular spaces. The point is not merely to enlarge the category of manifolds, but to preserve enough local differential structure for tangent-space and operator asymptotics to remain meaningful at singular boundary points.

In the Lorentzian setting, the corresponding object is formulated differently. A stratified submanifold $M$2 of dimension $M$3 is a finite union of disjoint smooth $M$4-dimensional submanifolds-with-boundary together with their common boundaries. A submanifold-with-kinks is such a stratified submanifold for which at each point at most two of the smooth pieces meet; equivalently, each codimension-1 boundary stratum is shared by exactly two smooth pieces. Thus there are no “Y-shaped” junctions, only “$M$5”–shaped edges. Its kink set $M$6 is the set of points at which $M$7 is not a smooth embedded submanifold, and $M$8 (Struyve et al., 2013).

These two constructions are related by their treatment of piecewise-smooth geometry, but they are not identical. A plausible implication is that “manifolds with kinks” should be read as a family of closely related technical notions whose exact meaning depends on the analytic problem under study.

2. Tangent structure and inward sectors

On a smooth manifold with kinks, the tangent space $M$9 at a point $(U,\phi)$0 is defined as the space of smooth derivations at $(U,\phi)$1. In local coordinates it is isomorphic to $(U,\phi)$2. The nontrivial additional datum is the inward sector, or strictly inward cone,

$(U,\phi)$3

This is the set of tangent vectors that point into $(U,\phi)$4 (Pal et al., 10 Jul 2025).

For a Lipschitz-CDD border chart $(U,\phi)$5 at $(U,\phi)$6, the inward sector agrees via $(U,\phi)$7 with the open feasible-direction cone of the Euclidean domain $(U,\phi)$8 at $(U,\phi)$9. In local coordinates,

$\phi:U\to\mathbb{R}^d$0

At a classical boundary point, $\phi:U\to\mathbb{R}^d$1 is a half-space. At an essential corner of depth $\phi:U\to\mathbb{R}^d$2, it is a $\phi:U\to\mathbb{R}^d$3-fold positive cone. At a cusp such as $\phi:U\to\mathbb{R}^d$4 at the origin, $\phi:U\to\mathbb{R}^d$5 is a single ray and has no interior (Pal et al., 10 Jul 2025).

This geometry is the organizing principle of the graph-Laplacian theory. The asymptotic behavior is determined by the inward sector of the tangent space. This suggests that, near a kink, the decisive local datum is not a single normal vector but a cone of admissible directions, together with its spherical averages.

3. Local models of singularity

The framework is designed to include several local models in a unified way. A smooth boundary point is locally of the form

$\phi:U\to\mathbb{R}^d$6

with $\phi:U\to\mathbb{R}^d$7, and $\phi:U\to\mathbb{R}^d$8 is a half-space. A corner point of depth $\phi:U\to\mathbb{R}^d$9 in $(U,\phi)$0 is locally $(U,\phi)$1, so $(U,\phi)$2. A higher-order corner in $(U,\phi)$3 is locally $(U,\phi)$4. A pyramid apex in $(U,\phi)$5 is locally the cone $(U,\phi)$6, which is a kink not covered by corners. A cusp may be modeled by $(U,\phi)$7 at the origin, where the inward sector collapses to a single ray (Pal et al., 10 Jul 2025).

These examples clarify a common misconception: manifolds with kinks are not limited to manifolds with corners. The class was introduced precisely to accommodate boundary singularities that are more general than the corner case, while still admitting a coherent tangent/inward-sector analysis.

The Lorentzian notion imposes a different restriction. A submanifold-with-kinks allows only two smooth pieces to meet at a singular stratum. More complicated junctions are excluded. By the Seeley extension theorem, any submanifold-with-kinks can be approximated arbitrarily well by a smooth submanifold, but this is not available for more complicated stratifications (Struyve et al., 2013).

4. Gaussian-kernel graph Laplacian on manifolds with kinks

Given i.i.d. samples $(U,\phi)$8, density $(U,\phi)$9, and bandwidth $\phi:\overline U\to\mathbb{R}^d$0, the unnormalized graph Laplacian at a test point $\phi:\overline U\to\mathbb{R}^d$1 is

$\phi:\overline U\to\mathbb{R}^d$2

Its expectation is

$\phi:\overline U\to\mathbb{R}^d$3

At a smooth interior point,

$\phi:\overline U\to\mathbb{R}^d$4

as $\phi:\overline U\to\mathbb{R}^d$5 (Pal et al., 10 Jul 2025).

At a boundary or kink point, the expansion depends on the geometry of $\phi:\overline U\to\mathbb{R}^d$6. For Lipschitz-CDD boundary or kink points,

$\phi:\overline U\to\mathbb{R}^d$7

where

$\phi:\overline U\to\mathbb{R}^d$8

is the average inward-unit-vector,

$\phi:\overline U\to\mathbb{R}^d$9

and

${\mathcal C}^0$0

The pointwise asymptotic theorem states that if ${\mathcal C}^0$1 is a ${\mathcal C}^0$2-dimensional Riemannian manifold with kinks, ${\mathcal C}^0$3, ${\mathcal C}^0$4, and ${\mathcal C}^0$5 is interior or Lipschitz-CDD border, then

${\mathcal C}^0$6

as ${\mathcal C}^0$7 (Pal et al., 10 Jul 2025).

The standard special cases are recovered directly from the inward-sector geometry. In the interior, ${\mathcal C}^0$8 is symmetric, so ${\mathcal C}^0$9 and one recovers ${\mathcal C}^k$0. On a smooth boundary, ${\mathcal C}^k$1 is a half-space and ${\mathcal C}^k$2 is the inner normal, so ${\mathcal C}^k$3 blows up like ${\mathcal C}^k$4. At a ${\mathcal C}^k$5-corner, one obtains a sum of ${\mathcal C}^k$6 normal-derivative terms.

5. Convergence theory and numerical validation

The pointwise asymptotic expansion has a stochastic counterpart. Choosing ${\mathcal C}^k$7 with

${\mathcal C}^k$8

guarantees in-probability convergence of ${\mathcal C}^k$9 to the deterministic limit; with slightly stronger tails, one also has almost-sure convergence (Pal et al., 10 Jul 2025).

The numerical study reported for the theory uses a 3D ball and a 3D cube. For the unit ball, $\psi\circ\phi^{-1}$0 points are sampled uniformly, and $\psi\circ\phi^{-1}$1 is evaluated at the center and on $\psi\circ\phi^{-1}$2. The unscaled $\psi\circ\phi^{-1}$3 converges to $\psi\circ\phi^{-1}$4 in the interior and blows up like $\psi\circ\phi^{-1}$5 on the boundary. The scaled quantity $\psi\circ\phi^{-1}$6 converges numerically to the predicted constants $\psi\circ\phi^{-1}$7 and $\psi\circ\phi^{-1}$8, respectively (Pal et al., 10 Jul 2025).

For the cube, one observes the hierarchy

$\psi\circ\phi^{-1}$9

at an interior point, face midpoint, edge midpoint, and vertex. This matches the number of active inward-normals. Empirical error rates in ${\mathcal C}^k$0 and ${\mathcal C}^k$1 agree with the ${\mathcal C}^k$2 remainder and the concentration bounds. Within this framework, the inward sector is the only datum needed to govern the leading-order and next-order behavior of Gaussian-kernel graph Laplacians (Pal et al., 10 Jul 2025).

6. Foliations-with-kinks in relativistic Bohmian mechanics

A foliation-with-kinks of a Lorentzian manifold ${\mathcal C}^k$3 is a decomposition

${\mathcal C}^k$4

such that each ${\mathcal C}^k$5 is a spacelike submanifold-with-kinks of codimension ${\mathcal C}^k$6, and the union of all kink sets

${\mathcal C}^k$7

is itself a stratified submanifold of codimension ${\mathcal C}^k$8. Two additional conditions are imposed: each ${\mathcal C}^k$9 is a Cauchy hypersurface, and at every point of $M$00, including points of the kink set, all tangent vectors are spacelike. In $M$01 space-time, $M$02 therefore has dimension $M$03 (Struyve et al., 2013).

For $M$04 non-interacting Dirac particles in Minkowski space with multi-time wave function

$M$05

the simultaneous-configuration set

$M$06

is a submanifold-with-kinks of dimension $M$07. On $M$08, one defines a probability current $M$09 as a smooth $M$10-form. On each smooth leaf $M$11, $M$12 induces the usual Bohm-Dirac current vector field $M$13 in coordinates $M$14. Away from the kink set, $M$15 is $M$16 and satisfies the continuity equation

$M$17

Because the leaves are only piecewise smooth, $M$18 generally has jump discontinuities along the coordinate image of the kink set (Struyve et al., 2013).

The key compatibility condition across the kink set is flux continuity: $M$19 where $M$20 is a unit normal to a regular piece of $M$21, and $M$22, $M$23 are the one-sided limits. Under standard assumptions—$M$24, $M$25, smoothness away from $M$26, one-sided smooth extensions up to $M$27, the continuity equation away from $M$28, and flux continuity on $M$29—every integral curve of $M$30 can be unambiguously continued across $M$31 except on a set of initial data of measure zero, and the density $M$32 is equivariant in time (Struyve et al., 2013).

For the hypersurface Bohm-Dirac model, the condition holds automatically. The current form $M$33 is divergence-free on $M$34 and does not depend on the foliation. A local calculation shows that on each side of the kink set the induced flux agrees with the pull-back of $M$35, and Stokes’ theorem forces equality of the normal fluxes on the two sides. As a consequence, the model has globally well-defined continuous world lines and preserves the $M$36-distribution on each leaf (Struyve et al., 2013).

7. Generic kinks, limitations, and conceptual significance

A notable application is the foliation determined by the condition

$M$37

Even when one starts from a smooth initial Cauchy surface, the integral leaves of this law develop generic piecewise-smooth corners: the foliation acquires kinks. The resulting foliation still satisfies the Cauchy property, the spacelike-tangent condition, and the requirement that its kink set be a stratified submanifold of codimension $M$38. The Bohm-Dirac model therefore remains well-posed and equivariant for the $M$39 foliation despite its generic kinks (Struyve et al., 2013).

The same paper also delineates a limitation. Guidance laws whose current depends explicitly on the normal $M$40 at each point, such as Slater’s photon law $M$41, will typically fail the flux-continuity condition at kinks and hence can lose equivariance or even trajectory continuation. Exactly at a single-particle crossing of $M$42, only the other particles’ velocities jump, because in the Bohm-Dirac velocity formula each particle’s velocity depends on the normals at the other particles’ positions (Struyve et al., 2013).

Taken together, the two research programs isolate a common theme. In the graph-Laplacian setting, manifolds with kinks provide a unifying framework for interior points, smooth boundary, corners, and more singular boundary-like structures, with the inward sector controlling the asymptotics. In the Bohmian setting, submanifolds-with-kinks are the “next-to-smooth” class of submanifolds: they capture generic singularities such as edges and corners while retaining a simple topological and measure-theoretic framework in which probability currents and ODE theory still apply. The phrase “manifolds with kinks” therefore denotes a mathematically controlled enlargement of smooth geometry, but the exact admissible singularities depend on the analytic role the kink is meant to play (Pal et al., 10 Jul 2025, Struyve et al., 2013).